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The most general Hamiltonian (in units of kT) is
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where K a is a coupling constant for the set of spins in Sa' and the sum over a extends over all possible sets la' Note that
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LE{s} =0
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In principle we can solve for K a through
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Ka =
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For formal manipulations it costs us nothing to regard K a as completely arbitrary. For practical purposes it suffices for us to think of E {s} as
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E{s} = K 1 L s;
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+ K2 L
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+ K3 L
(18.21 )
where < i) denote nearest-neighbor pairs, i, i next-to-nearest pairs, and i, < i, k) nearest-neighbor triplets, etc. i, Now divide the entire lattice into identical cubical blocks that cover the whole lattice, with b sites along each edge of the block. There are thus b d spins in the block B, which we denote collectively as {s} B' The block spin is
s~ =
f{ s} B
where f is a mapping of {s} B into the set {I, -I}. For example, for the majority rule with tiebreaker, f is the function defined in (18.4). It is convenient to define
where 13 K is the Kronecker 13. This function tells us whether a particular configuration {s} B gives S B= 1 or S B= - 1. Taking the product of PB over all blocks, we have a weight function
P{S',S} = TIP B
which depends on the set of all block spins {s'} and the set of all original spins {s}. It is equal to 1 if {s} gives rise to {s'}, and 0 otherwise. Clearly,
These are the only properties of a block spin we shall use. The partition function of the system can now be written as
e-E{s} =
L L P{s', s} e-E(s}
(Sf) (s)
where we have used (18.25). The block-spin Hamiltonian E' {s'} is defined by
L P{s', s} e-E{s}
where the constant Jl is to be so chosen that
E' {s'} = 0
which conforms to (18.19). Thus, E' is again of the form (18.18), except that K a is replaced by a new value K~. * We can now rewrite (18.24) as
The transformation from E to E' is called a renormalization-group (RG) transformation, formally indicated by
In the limit of an infinite system, the sets {s} and {s'} become the same, and only the coupling constants K a change in an RG transformation. Thus it is more appropriate to represent an RG transformation in the form
= Ra(K I , K 2 ,
Regarding K as components of a vector, and R a matrix operator, we can also write K' = R(K) (18.32) The RG transformations are so named because they" renormalize" the coupling constants, and that they have group property: If RI(K) and R 2 (K) are RG transformations, so is R I R 2 (K). They do not form a group because block spins cannot be "unblocked," and thus there are no inverse transformations. (Mathematically one can invert the map R, except possibly at isolated singular points; but this has no physical relevance.) Define the free energy per spin g(K) (in units of kT) by
L e-E{s}
In the block-spin system we have
e-N'g(K') =
{s' }
where N' = b-dN. The same function g appears in both (18.33) and (18.34) because E { s} and E'{s'} are the same functions except for the values of the
"The reason is that (18.18) is the most general form of the Hamiltonian whether we are dealing wi th the original spins or the block spins.
coupling constants. Using (18.29) we obtain
p.(K) + b-dg(K ' )
This, together with (18.32), describes how the system behaves under an RG transformation, which increases the unit of length by a factor b.
Let K(n) denote the coupling constants resulting from n successive applications of a given RG transformation. These coupling constants are given by the recursion relation (18.36)
It is important to note that R is independent of n, a fact that can be proven by constructing R formally through (18.22). A fixed point K * of the map R is defined by
We assume that K(n) approaches a fixed point as n ---) 00.* The Hamiltonian E * corresponding to K * is called the fixed-point Hamiltonian. A fixed point could be physically significant, because it is a point at which the system becomes invariant under a change of length scale. That means the correlation length is either 0 or 00. The latter corresponds to a critical point, which is the physically interesting case. The case with zero correlation length, as we have encountered in the one-dimensional Ising model, corresponds to infinite temperature, and usually can be recognized and rejected. We now investigate the behavior of the system near a fixed point, which we assume to correspond to a critical point. Subtracting (18.37) from (18.36), we have K(n+l) - K* = R(K(n ) - K* (18.38) Assuming n to be very large we can make the linear approximation
R(K(n )
R(K*) + W(K(n) - K*)